Rigid gauges and F-zips, and the fundamental sheaf of gauges G_1tnn [Elektronische Ressource] / vorgelegt von Felix Schnellinger
81 Pages
English
Downloading requires you to have access to the YouScribe library
Learn all about the services we offer

Rigid gauges and F-zips, and the fundamental sheaf of gauges G_1tnn [Elektronische Ressource] / vorgelegt von Felix Schnellinger

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer
81 Pages
English

Description

Rigid gauges and F-zips, and thefundamental sheaf of gauges GnDissertation zur Erlangung des Doktorgrades der Naturwissenschaften(Dr. rer. nat.) der mathematischen Fakultat¨ der Universit¨at Regensburgvorgelegt vonFelix Schnellinger aus Regensburg2009Promotionsgesuch eingereicht am: 30.01.2009Die Arbeit wurde angeleitet von Prof. Dr. Uwe JannsenPrufungsaussc¨ huss:Prof. Dr. Felix Finster (Vorsitzender)Prof. Dr. Uwe Jannsen (1. Gutachter)Prof. Dr. Torsten Wedhorn, Universit¨at Paderborn (2. Gutachter)Prof. Dr. Klaus Kunne¨ mannProf. Dr. Guido Kings (Ersatzprufer)¨IntroductionIn this paper we study D−ϕ-gauges introduced by J.M. Fontaine and U. Jannsen, theauthor’s advisor. These are objects of (Frobenius)-linear algebra over a perfect field kof positive characteristic. Fontaine and Jannsen define invariants for smooth projectivevarieties over k, which take values in gauges, by means of e.g. syntomic cohomology of a· crissheaf G built essentially of the sheavesO .n nIn the first section we study D −ϕ-gauges over a perfect field k. A D −ϕ−gauge1 1over k is a graded module M of finite type over the graded ring k[f,v]/(fv) (with fin degree 1 and v in degree−1) together with an Frobenius-semi-linear isomorphism∼∞ −∞ϕ :M →M . Fontaine defined the subcategory of rigid gauges to be those D −ϕ-1gauges with imv = kerf, imf = kerv and ker(f,v) = 0.WestudythestructureandmorphismsofrigidD−ϕ-gauges.

Subjects

Informations

Published by
Published 01 January 2009
Reads 7
Language English

Exrait

Rigid gauges and F-zips, and the
fundamental sheaf of gauges Gn
Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften
(Dr. rer. nat.) der mathematischen Fakultat¨ der Universit¨at Regensburg
vorgelegt von
Felix Schnellinger aus Regensburg
2009Promotionsgesuch eingereicht am: 30.01.2009
Die Arbeit wurde angeleitet von Prof. Dr. Uwe Jannsen
Prufungsaussc¨ huss:
Prof. Dr. Felix Finster (Vorsitzender)
Prof. Dr. Uwe Jannsen (1. Gutachter)
Prof. Dr. Torsten Wedhorn, Universit¨at Paderborn (2. Gutachter)
Prof. Dr. Klaus Kunne¨ mann
Prof. Dr. Guido Kings (Ersatzprufer)¨Introduction
In this paper we study D−ϕ-gauges introduced by J.M. Fontaine and U. Jannsen, the
author’s advisor. These are objects of (Frobenius)-linear algebra over a perfect field k
of positive characteristic. Fontaine and Jannsen define invariants for smooth projective
varieties over k, which take values in gauges, by means of e.g. syntomic cohomology of a
· crissheaf G built essentially of the sheavesO .n n
In the first section we study D −ϕ-gauges over a perfect field k. A D −ϕ−gauge1 1
over k is a graded module M of finite type over the graded ring k[f,v]/(fv) (with f
in degree 1 and v in degree−1) together with an Frobenius-semi-linear isomorphism
∼∞ −∞ϕ :M →M . Fontaine defined the subcategory of rigid gauges to be those D −ϕ-1
gauges with imv = kerf, imf = kerv and ker(f,v) = 0.
WestudythestructureandmorphismsofrigidD−ϕ-gauges. TheunderlyingD -module1 1
Ld
of a rigid gauge is isomorphic to D (m ) with some numbers m . The morphisms1 k kk=1
ofD −ϕ-gauges can be described by matrices overk which satisfy some Frobenius-linear1
equations. Thecompositionofmorphismsisingeneralnotgivenbymatrixmultiplication,
but we give an explicit description of the matrix of a composition of two morphisms.
AnF-zipoverk isafinite-dimensionalk-vectorspace, withanascendingandadescending
filtration with semi-linearily isomorphic subquotients (see A.2). There is a functor from
−∞rigid D −ϕ-gauges to F-zips over k, due to Fontaine, by sending M to M . The1
∞ ∞ ∞ r −∞filtrations are defined by the images ofv respϕf (With e.g. v the mapM →Mr r r
induced by v). We construct a functor in the opposite direction by mapping an F-zip
• σ r(M,C ,D ,ϕ) to (⊕ D )× D σ (⊕C ), which is a rigidD −ϕ-gauge. The main result• r gr ( M) 1
is that these functors are quasi-invers to each other, i.e. the category of F-zips over k is
a full subcategory of the category of D −ϕ-gauges. It is equivalent to the category of1
rigid D −ϕ-gauges.1
The second section introduces quasi-´etale morphisms. A morphism of schemes is called
quasi-´etale (or quiet) if is locally a composition of ´etale morphisms and successive ex-
ptractions of p.-th roots. By the latter we mean a morphism of rings A→A[T]/(T −α)
with some element α. Stability of quiet morphisms under compostition and base-change
is shown. One important property of classes of morphisms we study, is the ”lifting prop-
erty”: We say that a class τ of scheme morphisms satisfies the lifting property, if for
every nil-immersion U → T and for every τ-morphism f to U, Zariski-locally there is
a τ-morphism g to T, such that f is the base-change of g. It is shown that quasi-´etale
morphisms satisfy the lifting property.
Laterweshallshowthatthequasi-´etalecohomologyofcertainsheavesisequaltosyntomic
criscohomology. This is true for example forO .n
In the third section we will study different topologies and the associated cohomology for
direct images of quasi-coherent crystals. After a quick review of the large crystalline
1site endowed with different topologies, we define three axioms on classes τ of scheme-
morphisms (T1)-(T3). (T1) lists some standard properties, like flatness, stability under
base-change etc., (T2) is similar to the lifting property and (T3) demands that every
extraction of a p.-th root has to be in τ. If a class satisfies all three axioms we call it
p-crystalline.
If the first two are satisfied for a class τ, we can construct a morphism of topoi from the
large τ-crystalline topos to the large τ-topos v : (X/S) →X : The main reason is,CRIS,τ τ
that the lifting property together with flatness ensures, that aτ-covering can be lifted to
a τ-crystalline covering of any PD-thickening.
A crystal is a special, ”rigid” sheaf on the crystalline site, and it is called quasi-coherent
if it is defined by quasi-coherent modules. In the following, cohomology of direct images
crisof quasi-coherent sheaves under the morphisms v is studied. For exampleO is then
direct image of the crystalline structure-sheafO . The main result is the followingX/Wn
comparison theorem:
0If two classesτ andτ of morphisms satisfy the three axioms,τ-cohomology of the direct
0image of a quasi-coherent crystal agrees with its τ -cohomology. The proof combines
qthe facts that higher direct images R v F of a quasi-coherent crystal F vanish and that∗
crystalline cohomology of a quasi-coherent crystal is independent of the topology on the
crystalline site.
ThefourthsectionisdevotedtothestudyofthesheavesG . TheyaredefinedbyFontainen
and Jannsen and are one of the central constructions. We shall give and proof a small
formulaire of elementary properties of G .n
cris cris 0First the sheavesO are defined: O (Y) = H (Y/W ,O ) and a relation to an Y/Wn n cris n
divided power envelope of the pre-sheaf of Witt-vectors is given. It follows that there
crisis a Frobenius ϕ onO . The image of Frobenius for n = 1 is determined: There isn
crisa canonical monomorphismO ,→O , and the image of Frobenius is the image of this1
crismonomorphism. Furthermore there is an epimorphismO O. Both compositions1
equal the respective Frobenius.
We study the fundamental exact sequence
cris cris cris0→O →O →O → 0.n m+n m
r nThe graded sheaf of ringsG is defined by lettingG be the cokernel ofp -multiplicationn n
ϕr cris cris crisˆon G = ker(O →O →O ) for m≥n+r (This is independent of m). There arem m m r
global sections f and v of respective degree 1 and−1. We show ”strictness”, i.e. that
n nf v∞ −∞(f ,v ) is injective and some kind of rigidity, namely that the sequences G →G →n nr r
n nv f crisG and G →G →G are exact. There is a ringhomomorphism ϕ :G →O , whichn n n n n n
r r crisis, on G , informally be given by ”division by p ” after Frobenius onO . The imagesn m
crisF = imϕ inO define an ascending filtration.r r n
[1]
Now we consider characteristic p, i.e. n = 1. The kernel of FrobeniusJ is a divided1
[r]cris rpower-ideal inO , and the higher powers define a descending filtration F =J on1 1
2crisO . It is a result of Fontaine, that the subquotients of both filtrations are isomorphic,1
r r−1via the Cartier-isomorphism. Kernel and cokernel ofv :G →G are isomorphic toF ,rn n
r−1 r rwhile kernel and cokernel off :G →G are isomorphic toF . The second statementr n n
follows from the first with rigidity and Cartier-morphism. There is an exact sequence
0→G →G →G → 0.n m+n m
We wish to compare chomology of G for different topologies. One possible way to don
ˇthis is viap-good algebras and Cech-cohomology. A smoothk-algebra is called good, if it
admits a system of parameters, and a k-algebra is p-good, if it is the quotient of a good
p palgebraA, by a regular sequence (f ,...,f ). Every syntomic k-scheme can be covered1 r
byp-good algebras inp-topology and every syntomic covering can bep-refined (i.e. in the
topology generated by p.-th roots) to a covering consisting of p-good algebras in a very
particular form, a so called p-good covering. Fontaine computed the value of the sheaves
[r] [r] [r+1]crisO ,J andJ /J explicitely over p-good algebras.1 1 1 1
ˇWe are then able to show with Cech-cohomology computations that cohomology of the
[r] [r+1]
subquotientJ /J does not depend on the p-crystalline topology used. This easily1 1
implies that the cohomology of G is also independent of this choice.n
In the fifth chapter we study relations between F-zips and different notions of gauges
over a scheme X over F . The correct notion should be the one of ϕ−G -crystal. Ap 1
crisϕ−G -module is a graded G -module M plus an isomorphism Φ :O ⊗ M →1 1 Gn nϕ-
crisO ⊗ M. A ϕ−G -module which comes from the small Zariski-site is called aG 1n npr-
ϕ−G -crystal. IfX is a field, aϕ−G -crystal is the same as aD −ϕ-gauge. It turned1 1 1
outthatitisnecessarytomodifythenotionofF-zipslightly, fordetailsseetheappendix.
It seemed also, that modified F-zips are the right definition for extending the notion of
F-zip to a higher level, i.e. for definig F-zips over W .n
First we introduce the notions of D −ϕ-gauges over X. These are gradedO[f,v]/(fv)-1
∞ (p) −∞modules plus an isomorphism (M ) → M . Again there is a notion of strictness
and rigidity, given exactly as in the case of fields. If we want to compare with F-zips we
have to introduce a property of locally freeness: A D −ϕ-gauge is called locally free if1
all graded pieces are locally free and if kernel and cokernel of v and f are locally directr r
summands. There is a functor from modified F-zips to D −ϕ-gauges which induces an1
equivalence of categories between modified F-zips and rigid locally free D −ϕ-gauges.1
The functor and its quasi-inverse functor are essentially given as in the case of fields.
To a D −ϕ-gauge M we can assign a G −ϕ-module by tensoring: G ⊗ M is a1 1 1 D1
0 cris 0D −ϕ-crystal, with the morphism D → G which is given by D =O ,→O = G1 1 1 1 1 1
and f7→f, v7→v.
3Acknowledgment
I wish to thank my doctoral advisor Uwe Jannsen, who supported me with his intellect
and experience in various discussions, for the opportunity to write this thesis. He always
had an open ear for my problems and questions and was often able to give my thoughts
a new direction. Also i want to thank Jean-Marc Fontaine for an inspiring discussion
during his visit in Regensburg. Finally i thank my wife Linda Heiss for her support and
patience.
Conventions and notations
• For a set E we let ]E be the cardinality of E.
• The n.-th unit-matrix is denoted by E .n
• For an abelian Group A we let A be p-torsion: A ={a∈A|pa = 0}.p p
L
r• All gradings are indexed withZ. We write a graded object M as M = M .
r∈Z
• For a graded object M we and n∈ N we define the n.-th twist of M to be the
r n+rgraded object M(n) with M(n) =M
• If f : M → N is a morphism of graded modules of degree n, we write f forr
r n+rM →N .
• Descending filtrations are marked with an upper index, ascending filtrations with a
lower index.
•• Let C and D be a descending and an ascending filtration of an object M. If the•
subquotients exist we define
r D r r+1gr M =C /C and gr M =D /D .r r−1C r
• Monomorphisms are symbolized by ,→, epimorphisms by and for isomorphisms

we use→.
• If f : X→ Y is morphism of schemes, the associated morphismO → fO willY ∗ X
] −1 ∗be denoted by f . For anO -module M we denoteO ⊗ −1 f M with f M.Y X f OY
(p)• If M is aO -module over a scheme X/F we let M = M⊗ O with theX p O %F XX
pabsolute Frobenius F :x7→x .
• By ”DP” we mean divided powers. If an ideal in a ring is furnished with divided
powers, we denote by γ the n.-th divided power.n
4Contents
Contents
1 Elementary calculations on rigid modules and gauges 6
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Structure and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Connections with F-zips . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Quasi-´etale morphisms 22
2.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Lifting property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Cohomology of quasi-coherent crystals 25
3.1 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 A morphism of topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Direct images of crystals and their cohomology . . . . . . . . . . . . . . . . 34
4 Topologies and the sheaves G 39n
cris4.1 The ringO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39n
4.2 The fundamental gauges G . . . . . . . . . . . . . . . . . . . . . . . . . . 45n
4.3 p-good algebras and coverings . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 The comparison theorem for G . . . . . . . . . . . . . . . . . . . . . . . . 58n
5 ϕ-G -modules and crystals 63n
5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Connections with F-Zips II . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A Appendix 75
A.1 Syntomic morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.2 F-zips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.3 Some p-valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
References 78
51 Elementary calculations on rigid modules and gauges
1 Elementarycalculationsonrigidmodulesandgauges
Letk be a perfect field of characteristicp> 0, let its absolute Frobenius be denoted byσ.
nLetW =W(k) be the ring of Witt-vectors ofk, andW =W/p as usual. The Frobeniusn
morphism onW andW will also be denoted byσ. It raises each component to thep.-thn
power.
σFor a module M over W (resp. W ) we define ( M) to be the scalar restriction alongn
pFrobenius: Multiplication of an elementm by a scalarλ is by definitionλ m. Forn = 1,
σ ∼i.e. for a k-vectorspace we have the alternative description ( M) =M⊗ k.k% −1σ
(p)If M is W-module, we let M = M⊗ W be scalar extension along FrobeniusW%σ
(analogously for W -modules).n
Definition 1.0.1. A σ-linear map Φ : M→ N of modules over W (resp. W ) is a W-n
σlinear (resp. W -linear) map M→ ( N). Equivalently we can say that Φ is an additiven
map, such that for any element m∈ M and any scalar λ it holds Φ(λm) = σ(λ)Φ(m)
p(=λ Φ(m) for n = 1).
Remark 1.0.2. Letα :V→W beamapofk-vectorspaceswithrespectivebasesv ,...,v1 n
andw ,...,w . Let the matrix ofα with respect to these bases be denoted byA = (a ).1 m ij
σ σ σThe matrix of α : V→ W is given by
1
σ σ pA := ( a ) := (a ).ij ij
σIndeed the underlying map is the same, hence, if we denote scalar multiplication in W
by∗, we have:
m
X
σα(v ) = a wi ki k
k=1
m
1X
p= a ∗w .kki
k=1
1.1 Preliminaries
Remark 1.1.1. (i) We say that a graded module M over a graded ring R is of finite type
(or finitely generated) if there are homogenous elements m ...,m , such that their R-1 r
linear span is M. Equivalently M is of finite type if there is an epimorphism of graded
R-modules
r
M
R(i )Mk
k=1
where R(i ) is R with twisted grading.k
The notion of gauges was introduced by Fontaine and Jannsen. They found many notions
of gauges in different situations, this is the simplest:
61 Elementary calculations on rigid modules and gauges
Definition 1.1.2. (Due to Fontaine and Jannsen)
(i) Let D be the Z-graded, commutative ring D = W[f,v]/(fv−p) where f and v are
variables of degree 1 and−1 respectively.
(ii) A D-module is a graded module over D of finite type.
(iii) For a D-module M we let
−∞ k∼M = lim M =M/(v−1)M
−→
k∈Z,≥
∞ k∼M = lim M =M/(f−1)M
−→
k∈Z,≤
where the transition maps in the limit are given by multiplication with v resp. f.
For the maps into the limit we write
∞ r −∞v :M →Mr
∞ r ∞f :M →Mr
∞ −∞(iv) A D−ϕ-module is a D-module M with a σ-linear map ϕ :M →M .
(v) A D−ϕ-module is called a D−ϕ-gauge if ϕ is an isomorphism.
nDefinition 1.1.3. For a natural number n we let D =D/(p ) =W [f,v]/(fv−p).n n
Similarily we get the notion of D -modules, D −ϕ-modules and D −ϕ-gauges.n n n
L
rRemark 1.1.4. (i) We will view a D -module M = M as a diagram of W -modulesn nr
r−1 r r+1···M M M ···
r r−1 r r+1withW -linear mapsv :M →M andf :M →M , such that for everyr it holdsn r r
v f =p and f v =p.r r−1 r r+1
(ii) Since a D -module is assumed to be of finite type, multiplication by f is an isomor-n
phism for very large degree and multiplication by v is an isomorphism for very small
degree. This follows from simple calculations in graded modules:
Let M be the quotient of D (i )⊕...⊕D (i ) by the submodule generated by homoge-n 1 n s
snous elements m ,...,m of respective degrees d ,...,d . Then the graded piece M is1 r 1 r
(for s≥ max{d ,...,d ,−i ,...,−i ,0})1 r 1 s
s+i s+i s−d s−d1 r 1 rf W ⊕...⊕f W /(f m ,...,f m )n n 1 r
and multiplication by f is an isomorphism. We can treat v analogously.
(iii) Thus we can in effect represent a D -module by a finite diagram like in (i):n
a a+1 bM M ···M
for some integers a≤b.
71 Elementary calculations on rigid modules and gauges
There are many interesting subcategories in the category of D -modules, especially forn
n = 1,i.e. incharacteristicp. Weneedthefollowingnotionsof”strictness”and”rigidity”,
due to Fontaine. They are defined purely in terms of easy linear algebra, but give a
subcategory which allows us to compare gauges with other constructions of algebraic
geometry, especially Moonen and Wedhorns F-zips.
Definition 1.1.5. (due to Fontaine.) Let M be a D -module.1
r r−1 r+1(i) M is called strict if M →M ⊕M is injective for all r∈Z.
(ii)M iscalledrigid ifM isstrictanditholdsimv = kerf andkerf = imv(orequivalently
if imv = kerf and imf = kerv for all r).r+1 r r−1 r
1.2 Structure and morphisms
First we find, that for rigid modules the dimension of the homogenous parts does not
change with varying degree.
Lemma 1.2.1. Let M be a rigid D -module.1
r r+1(i) It holds dimM = dimM for all r∈Z.
(ii) One has rkf −rkf = rkv −rkv ≥ 0.r r−1 r r+1
(iii) Furthermore v| and f| are monomorphisms.r imv r imfr+1 r−1
Proof. (i) Consider the exact sequence
f vr−1 rr−1 r r−1M → M →M
which gives the exact sequence
r0→ imf →M → imv → 0.r−1 r
rThus we have rkf +rkv = dimM and analogously by the exact sequencer−1 r
v fr+1 rr+1 r r+1M → M →M
rwe get that rkf +rkv = dimM for all r.r r+1
r(ii) Let n = dimM . It is (because of the strictness)
n ≥ dimkerv +dimkerfr r
= n−rkv +n−rkfr r
= rkf +rkv −rkv +n−rkfr−1 r r r
= rkf −rkf +nr−1 r
and a similar calculation shows that this equals rkv −rkv +n.r+1 r
2(iii)Ifv m = 0itfollows(v(vm),f(vm)) = (0,pm) = 0andstrictnessimpliesvm = 0.
8